The Volume of the Giant Component of a Random Graph with Given Expected Degrees

We consider the random graph model $G(\mathbf{w})$ for a given expected degree sequence ${\mathbf w} =(w_1, w_2, \ldots, w_n)$. If the expected average degree is strictly greater than $1$, then almost surely the giant component in $G$ of $G({\mathbf w})$ has volume (i.e., sum of weights of vertices in the giant component) equal to $\lambda_0 {\rm Vol}(G) + O(\sqrt{n}\log^{3.5} n)$, where $\lambda_0$ is the unique nonzero root of the equation \[ \sum_{i=1}^n w_i e^{-w_i\lambda} = (1-\lambda) \sum_{i=1}^n w_i, \] and where ${\rm Vol}(G)=\sum_i w_i.$